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In this paper, we propose a cognitive transmission scheme with the best relay selection in a multiple-relay cognitive radio network to improve the secondary transmission performance while guaranteeing the quality-of-service (QoS) of the primary transmissions. We derive the closed-form expressions of outage probability for the secondary transmissions, called secondary outage probability, with the constraint of ensuring a target outage probability of primary transmissions (primary outage probability) for both the traditional non-relay and proposed best-relay selection based cognitive transmission schemes. Numerical results illustrate that, under a primary outage probability constraint, a secondary outage probability floor of the cognitive transmission occurs in high signal-to-noise ratio regions. Besides, the secondary outage probability floor of the proposed scheme is lower than that of the non-relay transmission, which is further reduced with an increasing number of cognitive relays.